to me, zuko’s relationship with toph is so antagonistic on his behalf because she’s the one member of the gaang (outside of suki who he simply doesn’t consider to be in the gaang tbh) who he least empathizes with, at large and in relation to her trauma. the reason for this is that unlike aang’s, katara’s, and sokka’s traumas which he sympathizes with + connects his trauma to in various ways of arguable aptness, zuko is uncomfortable with the striking closeness of toph’s trauma to his because of how class and privilege factor into their both of their trauma. aang, katara, and sokka are not privileged in any real way, esp not one that reminds him at all of his situation re status and privilege.
zuko finds aang, katara, and sokka's trauma relatable on a level that makes him feel equally victim of the fire nation; that's why he draws lines between kya being murdered in an ongoing genocide to protect katara and ursa, who was certainly not anti imperialism from anything we've seen of her, disappearing to protect him from his abusive father and grandfather. both situations are unpleasant and involve maternal sacrifice, but one is a domestic ordeal while the other is an act of racial violence and genocide. the connection exists, but the context is radically different. still, zuko likes that he can draw those parallels between his trauma and katara's. it allows him more room to feel like a better victim and arguably a bigger one in the context of the war, despite the fact that his status means he did in many ways benefit from his family and country being the perpetrators of it and his own personal role in it throughout books 1, 2, and 3.
i will give the obligatory disclaimer that zuko is a child soldier and he does side with the oppressed in the end, but he himself is not oppressed by the fire nation's genocidal war as a fire nation prince, let alone when he is the fire nation crown prince. he is someone who directly benefited from this war. yes, his father was abusive, yes, he was exiled for 2 and a half years, but he was still afforded many privileges because of the war. that does not negate his domestic suffering or his political exile, but those privileges still came about because of the suffering of others, including the gaang.
this is a nuanced situation. it is extremely complex and hard to accept the simultaneous truths at play here, even for adults, let alone for an extremely traumatized, black and white thinking 16 year old boy. zuko needs to be a victim in the context of the war as well as his family to alleviate his guilt about his participation in it prior to joining the gaang. he needs to relate his trauma to theirs. i will say that he doesn't ever seem to attempt this with suki nor does she offer her trauma to him, but that's more about both the writers and zuko not caring much if at all for her as a person.
but toph receives clear rejection from zuko when she tries to relate to him. her attempts to endear herself to him and to open up to him are shot down. why? surely toph suffered because of the war too. she's from the earth kingdom. but she never suffered a personal loss that zuko respected or related to. toph is not given any dead relatives. she was abused but not in a way zuko would understand, and her abuse did not negate that she lived a relatively comfortable life for being in the earth kingdom during the war. she was rich, she was cared for, she was shielded by privilege in many ways. while zuko is of course far richer than toph, he doesn't want to confront that reality. he doesn't want her privilege to make his obvious. he needs to focus on his suffering, his abusive father, his struggling to fit in as heir, and his time in exile and poverty. he can't accept the idea of those struggles coexisting with his privilege, so he can't accept the idea of toph's either. she has to just be whining about nothing. she has to be a nuisance not to be taken seriously. otherwise, he has to turn and face things about himself he doesn't want to be true.
could he grow to care about her and face his own privilege in the future? sure. but he doesn't in canon.
